André LeClair

LeClair's quantum field theory and quantum groups research have contributed to the identification of new types of quantum group symmetries. Focusing his research efforts on quantum group symmetries in two-dimensional quantum field theory, his work presented a systematic approach for expanding the symmetrical properties of a provided S-matrix[5] and introduced non-local conserved currents within the framework of two-dimensional quantum field theories, along with Denis Bernard.[6] In his examination of quantum affine symmetry,[7] he conducted a comprehensive examination of multiplicative representations of the Yangian double in collaboration with Bernard and Feodor Smirnov, and presented a geometric interpretation for the quantum double.[8] In related research, he investigated the S-matrices of integrable perturbations in N=2 superconformal field theories and obtained the Smatrix by employing the quantum group restriction of the affine Toda theories.[9]

LeClair's early research presented an examination of the sine-Gordon theory's Hilbert space and proposed a method to impose limitations on it while maintaining integrability by leveraging the quantum group structure. He also introduced fractional super soliton field theories, which are integrable and distinguished by a dual interplay of two fractional supersymmetries.[10] In a collaborative study with Giuseppe Mussardo and others, he developed a novel approach named "Rchannel TBA" to analyze the ground state energy of scalar integrable quantum field theories with boundaries.[11] His research on the integrability of coupled conformal field theories analyzed the Dynkin diagrams of coupled minimal models and provided a comprehensive conceptual structure for investigating the on-shell dynamics of interconnected conformal field theories. In addition, he proposed a comprehensive approach to the development of integrable defect theories through the application of perturbation techniques to conformal field theory and presented a formal framework for calculating correlation functions at finite temperatures in integrable quantum field theories which is commonly referred to as the LeClair-Mussardo formula.[12] More recently in 2022, his work suggested a systematic approach to classifying and understanding UV completions of 2D CFTs deformed by irrelevant TT perturbations and provided specific examples and results for the Ising model and other cases with different symmetries.[13]

LeClair's statistical mechanics research has had numerous implications for the field of theoretical physics.[14] He discovered the first examples of quantum field theories with cyclic renormalization group flows. In connection with this, he studied the Russian doll BCS model of superconductivity and established the presence of energetically elevated Cooper pairs, and possessed an interpretation within the framework of the renormalization group.[15] Dedicating his research efforts towards the statistical mechanics of gases, he presented multiple approaches, including the "Formalism" for quantum statistical mechanics of gases in any dimension, as well as the S-matrix approach to quantum gases in the unitary limit, particularly in two and three spatial dimensions.[16] In his assessment of the limitations and powers of the conformal bootstrap for two conformal theories, polymers and percolation, his study presented approximate numerical exponents, which, while reasonably accurate, fell short of the precision achieved by alternative methodologies like εexpansion or Monte-Carlo simulations.[17] While investigating the deformation of the Ising model and its ultraviolet completion, his study established that such deformations are generally incomplete in the UV due to square-root singularities in the ground state energy. Moreover, the study also proposed including an infinite number of additional irrelevant perturbations to complete the theory.[18] His study on the Ising model with a Majorana fermion spectrum revealed that a complete classification of integrable UV completions consists of only two possibilities, each with supersymmetry. He has also extended the Wigner-Dyson classification of random hamiltonians to non-hermitian hamiltonians, which recently have seen many applications to open quantum systems.

LeClair has proposed two approaches towards understanding the validity of the Riemann Hypothesis based on physics. One is based on the universal exponent 1/2 for random walks, which is exactly the real part of the non-trivial zeros.[19][20] In another approach he established the result that if there is a unique solution for every integer to the Franca-LeClair equation, then the Riemann Hypothesis is true. More recently he has constructed a physical model a fermion scattering with impurities whose quantized energies satisfy a Bethe ansatz equation which exactly correspond to the Riemann zeros.[21]